That makes us some kind of distant cousins.
In Watie's Regt, there is a Capt Richard Fields, age 39, that from your info I assume to be your Richard F. Fields b.1823.
There is also a "Rich" Fields shown with age 38 and "surgeon's interpreter". This may be your same Richard F.. This is from Grant Foreman's transcription and summary from the CMSRs and other sources so this could be the same person from two different sources.
Unfortunately, the name Richard is so common in the Fields family, its very difficult to keep them straight.
Capt Richard Fields of Drew's Regt (b. 1802) is the son of George Fields, my gggg grandfather. George's brother was Chief Richard Fields of the Texas Cherokee and their father is also Richard -- an Englishman who married Suzannah Emory.
I'm hoping you can help fill in the gaps. I don't know what Capt Richard Fields in the photo did after he was released from Camp Douglas POW camp in May 1862. I suspect he was helping Cherokee refugees in Texas and likely a refugee himself. It seems most likely that he was the Richard Fields who traveled to Richmond and corresponded with Lt Gen Kirby Smith and also in the photo with the post-war delegation to Washingtion. He died in Washington in 1873 is buried in the Congressional Cemetery with his son-in-law William Shorey Coody (d.1849).